Strong metrizability for closed operators and the semi-Fredholm operators between two Hilbert spaces

Abstract

To be able to refine the completion of C (H1, H2), the of set all closed densely defined linear operators between two Hilbert spaces H1 and H2, we define in this paper some new strictly stronger metrics than the gap metric g and we characterize the closure with respect to theses metrics of the subset L (H1, H2) of bounded elements of C (H1, H2). In addition, several operator norm inequalities concerning the equivalence of some metrics on L (H1, H2) are presented. We also establish the semi-Fredholmness and Fredholmness of unbounded in terms of bounded pure contractions.